TITLE: Why are the quaternions not an algebra over the complex numbers?
QUESTION [9 upvotes]: I just began to study about algebras over rings and quickly came across the fact that the quaternions are not an algebra over the complex numbers. I would prefer an answer as elementary as possible.

REPLY [4 votes]: Because the center of $\mathbb H$ is $\mathbb R$.

Why?
Suppose $a+bi+cj+dk\in Z(\mathbb H)$, the center of $\mathbb H$. Then $i(a+bi+cj+dk)=-b+ai-dj+ck$ and $(a+bi+cj+dk)i=-b+ai+dj-ck$ have to be equal, so $c=d=0$. Now, $(a+bi)j=aj+bk$ and $j(a+bi)=aj-bk$ so $b=0$. Thus, $Z(\mathbb H)\subseteq\mathbb R$, and the opposite inclusion is obvious.

Why does this suffice?
If $\mathbb H$ were a $\mathbb C$-algebra, then since $\mathbb C$ is commutative, we need $\mathbb C\subseteq Z(\mathbb H)$.
In general, if $A$ is a commutative ring, for any $A$-algebra $B$ we need $A\subseteq Z(B)$.